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Full text of "Analytical Mechanics"

IMPULSE AND MOMENTUM 247 velocity at that instant. During the compression part oi the collision a fraction of their energy is transformed into potential energy of compression and the rest into heat energy. During the restitution a fraction of the potential energy is transformed into kinetic energy and the rest into heat energy. Thus, in general, the kinetic energy at the end of the collision is less than that at the beginning. Therefore the relative velocity at the end of the collision is less than that at the beginning. Thus the coefficient of restitution is, in general, less than unity. If none of the energy, which is due to the relative motion of the colliding bodies, is lost in the form of heat, it is all transformed into potential energy during the compression and back into kinetic energy during the restitution. In this case the relative velocity at the end of the collision equals that at the beginning, which makes the coefficient of restitution unity.* The relative velocity at the end of the collision may be made greater than that at the beginning by having explosives at the point of contact. But this does not come in the definition of the coefficient of restitution. Therefore unity is the highest value of 6. When all the kinetic energy is transformed into heat during the collision the bodies have no. relative velocity after the collision. In this case the contact is called perfectly inelastic. Evidently e is zero when the contact is perfectly inelastic. Therefore the value of e lies between zero and unity. The values of the coefficient of restitution are 0.95 for glass on glass, 0.81 for ivory on ivory, and 0.15 for lead on lead. 200. Loss of Kinetic Energy of Colliding Bodies.—The kinetic energy of a system equals the kinetic energy due to the linear motion of the system with the velocity of its * In working out problems in which the contact is perfectly elastic instead of introducing the coefficient of restitution make use of the principle of the conservation of energy. The conservation of dynamical energy holds only when the contact is perfectly elastic. But the conservation of momentum and the conservation (general) of energy are true under all circumstances.